This discussion is specifically tailored for dumb UPS control.
However, most of the process is about the same for dumb UPSs and smart
UPSs. The biggest difference is in the details of how the UPS
monitoring daemon (typically powerd) communicates with the UPS.

Before doing anything, I suggest the following algorithm:

Skim this document.

Download and investigate all packages which seem specifically
tailored to your UPS.

Download and investigate the more generic packages. Note that
some of the more generic packages are actually more powerful,
better documented, and easier to use than their more specific
counterparts.

If you still can't get things working, or if points are still
unclear, read this document more carefully, and hack away...

When the power goes out, the UPS continues to power the computer and
signals that the power went out by throwing a relay or turning
on an opticoupler on it's control port.

Cable's job

The cable is designed so that when the UPS throws said relay,
this causes a particular serial port control line (typically
DCD) to go high.

Powerd's job

The powerd daemon monitors the serial port. Keeps raised/lowered
whatever serial port control lines the UPS needs to have
raised/lowered (typically, DTR must be kept high and whatever line
shuts off the UPS must be kept low). When powerd sees the
UPS control
line go high, it writes FAIL to /etc/powerstatus and
sends the
init process a SIGPWR signal. (Older versions of
powerd and initd wrote to /etc/powerfail.)
When the control line goes low again, it writes OK to
/etc/powerstatus and sends init
a SIGPWR signal.

Init's job (aside from everything else it does)

When it receives a SIGPWR, it looks at /etc/powerstatus.
If it contains FAIL it runs the powerfail entry from
/etc/inittab. If it contains OK it runs the
powerokwait entry from inittab.

If you need to make a cable, see section
How to make a cable for the overall details, and the
subsection of section
Info on selected UPSs that refers to your UPS. The latter might also include
information on manufacturer supplied cables. You may want to at least
skim all of section
Info on selected UPSs because each section has a few additional generally helpful
details.

Edit /etc/inittab. Put in something like this:

# What to do when power fails (Halt system & drain battery :):
pf::powerfail:/etc/powerfailscript +5
# If power is back before shutdown, cancel the running shutdown.
pg:0123456:powerokwait:/etc/powerokscript

Write scripts /etc/powerfailscript and
/etc/powerokscript to shutdown in 5 minutes (or whatever's
appropriate) and kill any existing shutdown, respectively.
Depending on the version of shutdown that you're using, this will
be either so trivial that you'll dispense with the scripts, or be a
1 line bash script, something along the lines of:

kill `ps -aux | grep "shutdown" | grep -v grep | awk '{print $2}'`

and you'll keep the scripts. (In case it doesn't come out right,
the first single quote on the above line is a backquote, the
second and third are single quotes, and the last is also a
backquote.)

Tell init to re-process the inittab file with the command:

telinit q

Edit rc.local so that powerd gets run upon startup. The syntax
is:

powerd <line>

Replace <line> with the serial port that the UPS
is connected, such as /dev/cua1.

Connect computer's serial port to UPS's serial port. DO NOT PLUG
THE COMPUTER INTO UPS YET.

Plug a light into the UPS.

Turn on the UPS and the light.

Run powerd.

Test the setup:

Yank the UPS's plug.

Check that the light stays on.

Check that /etc/powerfailscript runs.

Check that shutdown is running.

Plug the UPS back in.

Check that the light stays on.

Check that /etc/powerokscript runs.

Check that /etc/powerfailscript is not running.

Check that shutdown is no longer running.

Yank the UPS's plug again. Leave it out and make sure that the
computer shuts down properly in the proper amount of time.

The Dangerous Part. After everything seems to be
proper, power down the computer and plug it into the UPS. Run a
script that sync's the hard disk every second or so.
Simultaneously run a second script that keeps doing a find over
your entire hard disk. The first is to make this a little
safer and the second is to help draw lots of power. Now, pull
the plug on the UPS, check again that shutdown is running
and wait. Make sure that the computer shuts down cleanly before the
battery on the UPS gives out. This is dangerous because if the
power goes out before the computer shuts down, you can end up with a
corrupt file system, and maybe even lose all your files.
You'll probably want to do a full backup before this test, and
set the shutdown time extremely short to begin with.

Congratulations! You now have a Linux computer that's protected by a UPS
and will shutdown cleanly when the power goes out!